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# Homology, Homotopy and Applications

## Volume 5 (2003)

### Number 1

### Hochschild cohomology and moduli spaces of strongly homotopy associative algebras

Pages: 73 – 100

DOI: http://dx.doi.org/10.4310/HHA.2003.v5.n1.a5

#### Author

#### Abstract

Motivated by ideas from stable homotopy theory we study the space of strongly homotopy associative multiplications on a two-cell chain complex. In the simplest case this moduli space is isomorphic to the set of orbits of a group of invertible power series acting on a certain space. The Hochschild cohomology rings of resulting $A_{\infty}$-algebras have an interpretation as totally ramified extensions of discrete valuation rings. All $A_{\infty}$-algebras are supposed to be unital and we give a detailed analysis of unital structures which is of independent interest.

#### Keywords

$A_{\infty}$-algebra, derivation, Hochschild cohomology, formal power series

#### 2010 Mathematics Subject Classification

11S15, 13F25, 13N15, 14J10, 16E45